Analysis of a singly Reinforced beam

singly Reinforced beam

The beam that is longitudinally reinforced only in tension zone, it is known as singly reinforced beam. In Such beams, the ultimate bending moment and the tension due to bending are carried by the reinforcement, while the compression is carried by the concrete.

Practically, it is not possible to provide reinforcement only in the tension zone, because we need to tie the stirrups. Therefore two rebars are utilized in the compression zone to tie the stirrups and the rebars act as false members just for holding the stirrups.

Consider a singly reinforced beam section subjected to bending as shown in fig

stress and strain distribution in a singly reinforced beam as per Is 456.jpg

1.Strain Distribution:

The assumption (1) of the limit state theory gives a linear strain distribution across the cross section as shown in fig (b).

strain diagram

It varies as zero at the neutral axis and maximum at the extreme fibres.

2. Stress distribution

It has a parabolic shape from A to B and then linear from B to C above the neutral axis.The various salient points of the stress diagram are:

(i) Stress at neutral axis (Pt. A) = 0

(ii) Stress at 0.002 strain (Pt. B)

\[\frac{0.67f_{ck}}{1.5}= 0.446 f_{ck}\]

(iii) Stress at extreme fibre (Pt. C) = 0.446 fck

(iv) Below the neutral axis, the concrete is assumed to be cracked and maximum stress in steel

\[\frac{f_{y}}{1.15} = 0.87f_{y}\]

3. Stress block parameters

For the stress strain curve of concrete the design stress block parameter are taken as following as per IS 456:2000

Area of stress block = 0.36fckxu

Depth of centre of compressive force from the extreme fibre in compression = 0.42xu

concrete stress – block parameters in compression

4. Neutral axis depth ( xu )

Neutral axis is the axis at which the stresses are zero and it is situated at the centre of gravity of the section. The depth of neutral axis for a singly reinforced beam is calculated by equilibrium of tensile and compressive forces as shown in fig

Neutral axis depth

b = Width of section

d = Effective depth of beam

Ast = Area of steel reinforcement
xu = Depth of neutral axis
C = Total compression
T = Total tension
Total tension (T) = 0.87fy.Ast
Total compression (C) = 0.36 fckxu.b
at 0.42xu from the top extreme fibre (as per IS 456:2000)
For equilibrium of forces
Total Tension = Total compression
0.87fy.Ast = 0.36 fckxu.b
\[x_{u} = \frac{0.87f_{y}A_{st}}{0.36f_{ck}bd}\]
which is given in clause G1.1 of IS code
\[x_{u} = \frac{0.87f_{y}A_{st}}{0.36f_{ck}bd}\]

4.1 Limiting depth of neutral axis (xu max)

The strain distribution for the singly reinforced beam is shown in fig.

strain diagram

The maximum strain in concrete is 0.0035.
As per code, The strain in steel at failure should not be less than
\[\epsilon _{su}=\frac{0.87f_{y}}{E_{s}}+0.002\]
From the fig.
[latexpage]

\[\frac{0.0035}{x_{umax}}=\frac{\frac{0.87f_{y}}{E_{s}}+0.002}{d-x_{umax}}\]
\[\frac{x_{u max}}{d-x_{u max}}= \frac{0.0035}{\frac{0.87f_{y}}{E_{s}}+0.002}\]
\[x_{u max}\left ( \frac{0.87f_{y}}{E_{s}}+0.002+0.0035 \right )= 0.0035d\]
\[\frac{x_{u max}}{d}= \frac{0.0035}{\frac{0.87f_{y}}{E_{s}}+0.0055}\]
The above equation gives the limiting or maximum values of depth of neutral axis for different grades of steel.
The following table is based upon the above equation and gives limiting values depth of neutral axis for different grades of concrete.

Table Values of xumax/d
[table id=5 /]
The value of Es or modulus of elasticity is taken as 2×105N/mm2